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We classify the empirical literature on the wage impact of immigration into three groups, where studies in the first two groups estimate different relative effects, and studies in the third group estimate the total effect of immigration on wages. We interpret the estimates obtained from the different approaches through the lens of the canonical model to demonstrate that they are not comparable. We then relax two key assumptions in this literature, allowing for inelastic and heterogeneous labor supply elasticities of natives and the "downgrading" of immigrants. "Downgrading" occurs when the position of immigrants in the labor market is systematically lower than the position of natives with the same observed education and experience levels. Downgrading means that immigrants receive lower returns to the same measured skills than natives when these skills are acquired in their country of origin. We show that heterogeneous labor supply elasticities, if ignored, may complicate the interpretation of wage estimates, and particularly the interpretation of relative wage effects. Moreover, downgrading may lead to biased estimates in those approaches that estimate relative effects of immigration, but not in approaches that estimate total effects. We conclude that empirical models that estimate total effects not only answer important policy questions, but are also more robust to alternative assumptions than models that estimate relative effects.